Outline • What is mesoscale ? • Mesoscale statics and dynamics through coarse-graining. • Coarse-grained equations for a binary fluid-fluid mixture. • Numerical methods of solution. • Example simulations.
Numbers and methods 1 2 3 4 5 6 7 0 Ab initio methods Atomistic methods Continuum methods DFT CPMD MD LD BD DPD CFD LB Number of atoms (log) Simulation method Example of method
How lengths scale with numbers !! Try this for water Molar mass = 18 gm Molar volume = 18 mL N = 1023 L 107˚ A L N 1 3 N = 1 L 1˚ A !! Are other scalings possible ? How does length grow with size for a polymer ? Throughout, (!!) indicate exercises/derivations/points to ponder.
Length and time scales pico milli micro nano nano micro milli L T H = E m ¨ R = ⇥U ˙ R = U ˙ c = D 2c !! Are there systems with millimeter L but picosecond T ?
How times scale with masses and lengths m¨ x + kx = 0 2 0 = k m 0 m k ¨ u = c2 2u 0 = ±cq ⇥0 c ˙ u = D 2u i 0 = Dq2 ⇥0 2 D Harmonic Oscillator Wave Equation Diffusion Equation
Thermal fluctuations P(x |x) = 1 ⇤ 2 D⇥ exp (x x)2 2D⇥ ⇥ Brownian motion : Einstein (1905) D = kBT 6⇥ a temperature size Stokes-Einstein-Sutherland Relation Q. Why does a smaller particle have a higher diffusion coefficient ? A. The ‘root N’ effect! The central limit theorem helps us understand this. !! Is it (central limit) theorem or central (limit theorem) ?
Langevin theory inertia damping reversible force fluctuation deterministic stochastic mean behaviour ‘root N’ fluctuations regression to the mean equilibrium fluctuations v2⇥ = kBT F = 0 Free Brownian particle m˙ v + v F(x) = ⇥(t)
Coarse-graining in degrees of freedom What is the idea ? x1 x2 } P(x1, x2 ) RV y = x1 + x2 “Coarse-grained” sum P(y) P(y) = dx1dx2 (y x1 x2 )P(x1, x2 ) What is the distribution ? contains less information about the system than P(y) P(x1, x2 ) Adequate if we are only interested in the sum variable.
Coarse-grained Landau-Ginzburg functional Microscopic Hamiltonian Gibbs distribution P(q) = exp[ H(q)] Z H(q) = f(q) Order parameter Definition of Landau functional P(⇤) = ⇥ N i=1 dqi⇥[⇤ f(q)] exp[ H(q) Z ⇥ exp[ L] Z P(⇥) ⇥ exp[ L] Z
Or ansatz based on symmetry. L = A 2 2 + B 4 4 + K 2 (⇥ )2 P(⇥) ⇥ exp[ (A 2 ⇥2 + B 4 ⇥4)] P(s) ⇥ a+ (s 1) + a (s + 1) -J +J local part non-local part K( )2 All equal-time equilibrium properties are determined by this functional. !! Why is this called a “free energy” ? Is it a thermodynamic free energy ?
What about dynamics ? Temporal coarse-graining D = kBT 6⇥ a Why is this independent of particle mass ? ⇥d = m Inertial time scale. Time the velocity ‘remembers’ its initial condition. d ˙ x = v ˙ x = ⇥(t) Overdamped equations of motion. Valid when d m˙ v + v = ⇤(t) = 6⌅⇥a
Conserved order parameters Ising spin Lattice gas Ising spins are usually non-conserved d dt (r, t) = 0 Lattice gas models are conserved d dt (r, t) = 0
Numerical solution Stochastic PDE Stochastic ODE Stochastic realisation Spatial discretisation Temporal integration ⇤t⇥ = D⇤2 x ⇥ + ⇥x = (x + h) (x) h ⇤t⇥(i) = DL2 ij ⇥(j) + (i) ⇥(i, t + t) ⇥(i, t) = t+ t DL2 ij ⇥(j) + (i) !! Obtain the explicit form of the L matrix for a central difference Laplacian
Conclusion • Mesoscale methods are appropriate at length scales intermediate between the molecular and continuum scales. • The continuum description must be supplemented by fluctuation terms. At the mesoscale, the number of particles is not so large that fluctuations can be neglected. • Lengths and times must be coarse-grained. Intelligent coarse-graining improves computational efficiency, often by orders of magnitude, compared to direct MD. • Mesoscale methods can be particle-based, for instance dissipative particle dynamics and Brownian dynamics, or field-based like time-dependent Ginzburg-Landau theory and fluctuating hydrodynamics. • TDGL equations can be solved very efficiently on computers using matrix formulations.
Further reading • Stochastic processes in Physics and Chemistry : van Kampen • Handbook of stochastic processes : Gardiner • Principles of Condensed Matter Physics : Chaikin and Lubensky • Modern Theory of Critical Phenomena : Ma • Reviews of Modern Physics article : Halperin and Hohenberg • Numerical Solutions of Stochastic Differential Equations : Kloeden and Platen.
ronojoy
ssii
icoxfog417
eureco
toskamiya
nkmr_rl
ae14watanabe
yuyamaguchi
aronwalsh
fullfull
s1ok69oo
konakalab
ufuk
morganepeng
scottboms
bluesmoon
kneath
aleyda
sachag
garrettdimon
lara
danielanewman
samlambert
trishagee